Learn slope graphs
Slope measures a line's vertical change compared with its horizontal change. In slope-intercept form, y = mx + b, m is the slope and b is the y-intercept.
You will find slope from a graph or two points and build a line equation from a graph.
The idea in plain English
Slope tells how steep a line is and which way it moves. It compares vertical change with horizontal change.
Rise is the change in y. Run is the change in x. Slope is rise divided by run.
To turn a graph into an equation, find b where the line crosses the y-axis, find m with rise over run, then put both into y = mx + b.
How to recognise this kind of question
- You see a line, coordinate grid, or two ordered pairs.
- The question asks for slope, rate of change, rise, or run.
- You need to compare how y changes when x changes.
- If the question asks for the equation of a graphed line, you need both m and b.
Quick decision: Find b at the y-axis, find m with change in y over change in x, then build y = mx + b when the question asks for the equation.
The method
- Choose two clear points on the line and count the vertical change, or rise.
- Count the horizontal change, or run, moving in the same direction.
- Write slope as rise over run, simplify it, and use the y-intercept to check the line.
Worked examples
A line passes through (0, 1) and (2, 5).
- Rise: 5 − 1 = 4.
Answer: m = 2
Find the slope through (1, 5) and (4, −1).
- Change in y: −1 − 5 = −6.
- Change in x: 4 − 1 = 3.
Answer: m = −2
A line crosses the y-axis at 4 and the x-axis at −2. Find its equation.
- The y-intercept is 4, so b = 4.
- Move from (0, 4) to (−2, 0): rise = −4 and run = −2.
- m = −4/−2 = 2.
- Put m = 2 and b = 4 into y = mx + b.
Answer: y = 2x + 4
Your turn—with help
Find the slope through (−1, 2) and (3, 10).
Need a hint?
Subtract the coordinates in the same order.
m = (10 − 2)/(3 − (−1))
Show the answer and steps
- Rise: 10 − 2 = 8.
- Run: 3 − (−1) = 4.
Answer: m = 2
Watch out for this mistake
Incorrect: Slope = run/rise
Why it fails: The slope definition compares vertical change to horizontal change, not the reverse.
Do this instead: Use rise/run, or change in y/change in x.
Quick check
Try these without looking. Then open each answer to check your thinking.
Rise 6, run 3
Need a hint?
Slope is rise over run.
Show the answer and steps
Answer: m = 2
Through (0, 4) and (2, 0)
Need a hint?
Find both changes in the same order.
Show the answer and steps
- Rise: 0 − 4 = −4.
- Run: 2 − 0 = 2.
Answer: m = −2
A horizontal line
Need a hint?
Its rise is zero.
Show the answer and steps
- Rise/run = 0/run = 0.
Answer: m = 0
A line has m = −2 and crosses the y-axis at 3
Need a hint?
Put m and b into y = mx + b.
Show the answer and steps
- b = 3.
- m = −2.
- y = −2x + 3.
Answer: y = −2x + 3
Common mistakes
- Counting rise and run in opposite directions.
- Writing run over rise instead of rise over run.
- Plotting the y-intercept on the x-axis.
Practice tips
- Mark the two points used for each slope triangle.
- Check that positive lines rise left to right and negative lines fall left to right.